We present a methodology for coupling several saturation-based theoremprovers (running on different computers). The methodology is well-suited for re-alizing cooperation between different incarnations of one basic prover. Moreover,also different heterogeneous provers - that differ from each other in the calculusand in the heuristic they employ - can be coupled. Cooperation between the dif-ferent provers is achieved by periodically interchanging clauses which are selectedby so-called referees. We present theoretic results regarding the completeness ofthe system of cooperating provers as well as describe concrete heuristics for de-signing referees. Furthermore, we report on two experimental studies performedwith homogeneous and heterogeneous provers in the areas superposition and un-failing completion. The results reveal that the occurring synergetic effects leadto a significant improvement of performance.

In this paper we present an extensional higher-order resolution calculus that iscomplete relative to Henkin model semantics. The treatment of the extensionality princi-ples - necessary for the completeness result - by specialized (goal-directed) inference rulesis of practical applicability, as an implentation of the calculus in the Leo-System shows.Furthermore, we prove the long-standing conjecture, that it is sufficient to restrict the orderof primitive substitutions to the order of input formulae.

Proof planning is an alternative methodology to classical automated theorem prov-ing based on exhausitve search that was first introduced by Bundy [8]. The goal ofthis paper is to extend the current realm of proof planning to cope with genuinelymathematical problems such as the well-known limit theorems first investigated for au-tomated theorem proving by Bledsoe. The report presents a general methodology andcontains ideas that are new for proof planning and theorem proving, most importantlyideas for search control and for the integration of domain knowledge into a general proofplanning framework. We extend proof planning by employing explicit control-rules andsupermethods. We combine proof planning with constraint solving. Experiments showthe influence of these mechanisms on the performance of a proof planner. For instance,the proofs of LIM+ and LIM* have been automatically proof planned in the extendedproof planner OMEGA.In a general proof planning framework we rationally reconstruct the proofs of limittheorems for real numbers (IR) that were first computed by the special-purpose programreported in [6]. Compared with this program, the rational reconstruction has severaladvantages: It relies on a general-purpose problem solver; it provides high-level, hi-erarchical representations of proofs that can be expanded to checkable ND-proofs; itemploys declarative contol knowledge that is modularly organized.

We examine different possibilities of coupling saturation-based theorem pro-vers by exchanging positive/negative information. We discuss which positive ornegative information is well-suited for cooperative theorem proving and show inan abstract way how this information can be used. Based on this study, we in-troduce a basic model for cooperative theorem proving. We present theoreticalresults regarding the exchange of positive/negative information as well as practi-cal methods and heuristics that allow for a gain of efficiency in comparison withsequential provers. Finally, we report on experimental studies conducted in theareas condensed detachment, unfailing completion, and superposition.The author was supported by the Deutsche Forschungsgemeinschaft (DFG).

This report is a first attempt of formalizing the diagonalization proof technique.We give a strategy how to systematically construct diagonalization proofs: (i) findingan indexing relation, (ii) constructing a diagonal element, and (iii) making the implicitcontradiction of the diagonal element explicit. We suggest a declarative representationof the strategy and describe how it can be realized in a proof planning environment.

This paper presents a new kind of abstraction, which has been developed for the purpose of proofplanning. The basic idea of this paper is to abstract a given theorem and to find an abstractproof of it. Once an abstract proof has been found, this proof has to be refined to a real proofof the original theorem. We present a goal oriented abstraction for the purpose of equality proofplanning, which is parameterized by common parts of the left- and right-hand sides of the givenequality. Therefore, this abstraction technique provides an abstract equality problem which ismore adequate than those generated by the abstractions known so far. The presented abstractionalso supports the heuristic search process based on the difference reduction paradigm. We give aformal definition of the abstract space including the objects and their manipulation. Furthermore,we prove some properties in order to allow an efficient implementation of the presented abstraction.

We present a distributed system, Dott, for approximately solving the Trav-eling Salesman Problem (TSP) based on the Teamwork method. So-calledexperts and specialists work independently and in parallel for given time pe-riods. For TSP, specialists are tour construction algorithms and experts usemodified genetic algorithms in which after each application of a genetic operatorthe resulting tour is locally optimized before it is added to the population. Aftera given time period the work of each expert and specialist is judged by a referee.A new start population, including selected individuals from each expert and spe-cialist, is generated by the supervisor, based on the judgments of the referees.Our system is able to find better tours than each of the experts or specialistsworking alone. Also results comparable to those of single runs can be found muchfaster by a team.

We present a cooperation concept for automated theorem provers that isbased on a periodical interchange of selected results between several incarnationsof a prover. These incarnations differ from each other in the search heuristic theyemploy for guiding the search of the prover. Depending on the strengths' andweaknesses of these heuristics different knowledge and different communicationstructures are used for selecting the results to interchange.Our concept is easy to implement and can easily be integrated into alreadyexisting theorem provers. Moreover, the resulting cooperation allows the dis-tributed system to find proofs much faster than single heuristics working alone.We substantiate these claims by two case studies: experiments with the DiCoDesystem that is based on the condensed detachment rule and experiments with theSPASS system, a prover for first order logic with equality based on the super-position calculus. Both case studies show the improvements by our cooperationconcept.

This paper provides a description of PLATIN. With PLATIN we present an imple-mented system for planning inductive theorem proofs in equational theories that arebased on rewrite methods. We provide a survey of the underlying architecture ofPLATIN and then concentrate on details and experiences of the current implementa-tion.

We present an empirical study of mathematical proofs by diagonalization, the aim istheir mechanization based on proof planning techniques. We show that these proofs canbe constructed according to a strategy that (i) finds an indexing relation, (ii) constructsa diagonal element, and (iii) makes the implicit contradiction of the diagonal elementexplicit. Moreover we suggest how diagonal elements can be represented.